Naive, Biased, yet Bayesian: Can Juries Interpret Selectively ...

Can Juries Interpret Setecttajry Produced Evidence? 267
production, and therefore flips more in equilibrium. The equal marginal costs
of flipping, as opposed to the unequal costs of evidence production, can be
justified by assuming a competitive market for expert coin-flippers. An expert
charging a supracompetitive rate would not attract any clients.
To illustrate the equilibrium, we present a numerical example. Suppose that
the true p equals |, so that the plaintiff is favored by the distribution, and the
jury has a uniform prior over the unknown p, that is, [a = 1, b — 1}. Suppose
further that the amount at risk is $2,000,000, and the cost of flipping is $ 100,000.
In equilibrium the plaintiff reports 1.96 heads, and the defendant reports 0.48
tails. On average, the plaintiff will take 2.94 flips, costing $294,000, and
the defendant 1.44 flips, costing $144,000. The jury's estimate of p is exact:
(1 + 1.96)/(2 + 1.96 + 0.48) = .667.
When p is close to zero, or one, the marginal benefit of flipping is so small
that the party favored by the distribution takes no flips. In this case, the Nash
equilibrium takes the following form, depending upon who decides not to flip:
{//'*, T**) = {-a-b + (l/c)(-ac(l - p)) l/2 ,0) if 7"* < 0 (4b)
= {0, -a - b + (\/c)(bcpS) l/2 } if H* < O.(4c)
The stronger the beliefs of the jury—that is, the larger are {a, b]—the fewer flips
the respective parties take. A party produces no evidence when the bias of the
jury is strong, and when p is near zero or one. In this case the marginal benefit
of flipping is so small, that it is better to not flip at all. When this happens,
the opposing litigant decides to flip more because the reaction functions of the
litigants are upward sloping. The net effect of these changes is to bias the
estimator of the jury toward the prior mean of the jury. Note that although
the estimator is biased, it represents a improvement relative to the no-evidence
case, because it moves the jury's estimator from the prior mean toward the
true p.
The Nash equilibrium is illustrated for all possible p values in three different
cases for S = $2,000,000 and c = $100,000: in Figure 1, [a = 1, b = 1);
Figure 2, {a = 2, b = 1}; and Figure 3, {a = 0, b = 0}. In all three figures,
the true value of p is on the horizontal axis; the top graph plots the number of
heads and tails reported by the litigants; and the bottom graph plots the value
of the jury's estimator relative to the true value of p. In Figure 1, the jury has a
uniform prior over the unknown value of p. In the top graph of Figure 1, when
the value of p is near zero, only the plaintiff flips, and the jury's estimator of
p is biased toward the prior mean. The jury's prior is represented by a dashed
horizontal line. The bias is represented in the bottom graph as a deviation from
the 45 degree line. As p increases, when the defendant begins flipping, the
jury's estimator becomes exact—that is, p* — H*/(H* + T*) = p. As p
approaches one, the plaintiff stops flipping, and the jury's estimator is again
biased toward the prior mean.
The effects of jury bias are illustrated in Figure 2, where the jury's prior